5.1 Composite Functions
Explain the meaning of the following notation:
(f@g)(x)
(f@g)(x) = f(g(x))
means to put the g(x) function inside the f(x) function
Which line test helps determine if a graph is a one-to-one function or not?
The Horizontal Line Test!
How can you find the inverse of a function?
Swap x and y!
Then isolate the "new y" that you just swapped!
What is the y-intercept for the following exponential function?
f(x) = 5(2^x)
(0,5)
the a value is the y-intercept!
If a compound interest account is compounded quarterly, what is the n value for the formula?
n=4
f(x) = x^2-x+4
g(x) = 2x+5
find
(f@g)(-1)
(f@g)(-1)=10
Define a one-to-one function (without looking back in your notes!)
A one-to-one function is where any two different inputs (x-values) go to two different outputs (y-values)
What line does a graph and its inverse reflect over?
Evaluate the function when x=3
y=-1/4*2^x
y=-2
Paul invests $10,000 in a savings account with a fixed annual interest rate of 8% compounded annually. What will the account balance be after 5 years?
14693.28
f(x) = x^2-x-20
g(x) = -2x+7
find
(g@f)(x)
(g@f)(x) = -2x^2+x+40
Which is a 1-1 function? (Draw the graphs)
Option B!
How can you algebraically determine if f(x) and g(x) are inverse functions of one another?
Check if the following is true, if yes then they're inverses:
f(g(x)) = x = g(f(x))
Solve for x:
2^-x=16
x=-4
Nic invests $8,050 in a savings account with a fixed annual interest rate of 6.67% compounded monthly. What will the account balance be after 7 years?
12823.50
f(x) = x^2-x-20
g(x) = -2x+7
find
(f@g)(x)
(f@g)(x) = 4x^2-26x+22 = 2(2x^2-13x+11)
Is the inverse of the following set of ordered pairs a function:
{(-1,4) , (2,1) , (0,9) , (-9,0) , (5,-5)}
Yes!
No different x-values go to the same y-value so it's inverse will be a function!
{(4,-1) , (1,2) , (9,0) , (0,-9) , (-5,5)}
Find the inverse of the following function:
f(x) = -2x+5
f^-1(x) = -1/2x+5/2
Solve for x:
(e^x)^3=e^2e^-x
x=1/2
Rohan invests $9,047 in a savings account with a fixed annual interest rate of 11% compounded continuously. What will the account balance be after 10 years?
27178.69
Find the domain of f(g(x)) for the following functions.
f(x) = 1/(x-3
g(x) = 2/(x+1)
xne-1 and xne-1/3
If the function f(x) is a 1-1 function and f(3) = 8 then what is...
f^-1(8)
3
Find the inverse of the following function:
f(x) = -3/(x+2)
f^-1(x) = (-2x-3)/x
if
9^x=36
what does
3^-x "equal"
3^-x=1/6
Jon wants $80,000 in a savings account after 22 years. The account has an interest rate of 5.5% compounded continuously. What should Jon initially invest to obtain his goal?
23855.78
Find two components for the following function that is already composed as f(g(x))
1/sqrt(x+5
Lots of answers! (two main solutions below)
f(x) = 1/x and g(x) = sqrt(x+5
f(x) = 1/sqrt(x) and g(x) = x+5
f(x) is a 1-1 function. The domain of f(x) is:
[0,oo)
the range of f(x) is
(-oo,2) cup (5,7)cup (7,oo)
"find the domain and range of" f^-1(x)
Domain:
(-oo,2) cup (5,7)cup (7,oo)
Range:
[0,oo)
Find the inverse of the following function:
f(x) = (-1+x)/(x+5)
f^-1(x) = (5x+1)/(-x+1)
Solve for x:
5^(2x)*125^(x^2)=1/5
x=-1 and x=-1/3
Gabby wants $33,333 in a savings account after 6 years. The account has an interest rate of 9.5% compounded weekly. What should Gabby initially invest to obtain her goal?
18860.47